\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 33, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/33\hfil Blow-up solutions] {Integrability of blow-up solutions to some non-linear differential equations} \author[Michael Karls \& Ahmed Mohammed\hfil EJDE-2004/33\hfilneg] {Michael Karls \& Ahmed Mohammed} % in alphabetical order \address{Michael Karls \hfill\break Department of Mathematical Sciences\\ Ball State University\\ Muncie, IN 47306, USA} \email{mkarls@bsu.edu} \address{Ahmed Mohammed \hfill\break Department of Mathematical Sciences\\ Ball State University\\ Muncie, IN 47306, USA} \email{amohammed@bsu.edu} \date{} \thanks{Submitted December 25, 2003. Published March 8, 2004.} \subjclass{34C11, 34B15, 35J65} \keywords{Blow-up solution, Keller-Osserman condition, integrability} \begin{abstract} We investigate the integrability of solutions to the boundary blow-up problem $$r^{-\lambda}\bigl(r^{\lambda}(u')^{p-1}\bigr)'=H(r,u),\quad u'(0)\geq 0,\quad u(R)=\infty$$ under some appropriate conditions on the non-linearity $H$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{rem}[thm]{Remark} \newtheorem{example}[thm]{Example} \section{Introduction} Let $\lambda\ge 0$, $p>1$, $R>0$. For $00$ for all $s>0$. \end{itemize} Further assumptions on $H$ will be given as needed. In the literature, solutions of (\ref{meqnre}) are known as blow-up solutions, explosive solutions or large solutions. These type of equations arise as radial solutions of the $p$-Laplace equation, as well as the Monge Amp\'{e}re equation on balls. Radial solutions $u$ of the $p$-Laplace equation $$\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)=J(|x|,u),$$ in the ball $B:=B(0,R)\subseteq \mathbb R^N$ satisfy the first equation of (\ref{meqnre}) with $\lambda=N-1$, $H(r,u)=J(r,u)$. Likewise radial solutions of the Monge Amp\'{e}re equation $$\det (D^2u)=J(|x|,u),$$ in the ball $B$ also satisfy the first equation of (\ref{meqnre}) with $\lambda=0$, $p=N+1$ and $H(r,u)=Nr^{N-1}J(r,u)$. Noting that $u'$ is non-negative for any solution $u$ of (\ref{meqnre}), we will find it convenient to rewrite equation (\ref{meqnre}) as $$\label{meqn} \begin{gathered} \bigl((u')^{p-1}\bigr)'+\frac{\lambda}{r}(u')^{p-1}=H(r,u),\\ u(0)\geq0,\quad u'(0)\geq 0,\quad u(R)=\infty. \end{gathered}$$ A necessary and sufficient condition for the existence of a solution to problem (\ref{meqnre}) with $u'(0)=0$, $H(r,u)=f(u)$, and $f(0)=0$, is the (generalized) Keller-Osserman condition (see \cite{KEL,OSS,MAT}). $$\label{ok} \int_1^\infty\frac{ds}{F(s)^{1/p}}<\infty,\quad F(s)=\int_0^sf(t)dt.$$ If a nonnegative, non-decreasing continuous function $F$ defined on $[0,\infty)$ satisfies the Keller-Osserman condition (\ref{ok}) for some $p>1$, we will indicate this by writing $F\in KO(p)$. When $H(r,s)=f(s)$, and $\lambda=N-1$, problem (\ref{meqnre}) has been studied extensively by several authors, (see \cite{BAM, BAM3,KEL, LAI, LMK, MAT, OSS} and the references therein). The questions of existence, uniqueness and asymptotic boundary estimates have received particular attention. The case when $p=2$ and $H(r,s)=g(r)f(s)$ with $g\in C([0,R])$, possibly vanishing on a set of positive measure, has been considered in \cite{LAI}. In all these cases, the Keller-Osserman condition on $f$ remains the key condition for the existence of solutions. However, if $g$ is allowed to be unbounded the situation is completely different and existence and boundary behavior of a blow-up solution depends on how fast $g$ is allowed to grow near $R$. For such cases we refer the reader to \cite{MOH} or \cite{MPP}. For a discussion on solutions of (\ref{meqnre}) for general non-linearity $H$, we refer the reader to the paper \cite{RAW}. In this paper we are interested in studying the integrability property of blow-up solutions to (\ref{meqnre}) for $F\in KO(p)$. A blow-up solution may not have any integrability property at all, as the following example, taken from \cite{MOP}, shows. \begin{example} \label{ex1.1} \rm Let $u(r)=-1+e^{(1-r)^{-1}}$. Then \begin{gather*} u''(r)=f(u),\quad 00$. The antiderivative$F$of$f$that vanishes at zero is given by$F(s)=((s+1)^2\log^4(s+1))/2$, and observe that$F\in KO(2)$, but$F\notin KO(\alpha)$for any$\alpha>2.$\end{example} On the other extreme any positive power of a blow-up solution could be integrable. This can be seen from the following example. \begin{example} \label{ex1.2} \rm We fix$00$. In this example the primitive$F$of$f$with$F(0)=0$satisfies$F\in KO(\alpha)$for all$\alpha>0$. \end{example} The outline of the paper is as follows. In Section 2 we compare solutions$u$of (\ref{meqn}) with solutions of $$\label{seqn} \begin{gathered} ((w')^{p-1})'+\frac{\lambda}{r}(w')^{p-1}=H(0,w),\\ w(0)\geq 0,\quad w'(0)=0,\quad w(R)=\infty, \end{gathered}$$ for$0p$. Then$u\in L^{(\alpha-p)/p}(0,R)$for any solution$u$of (\ref{meqn}). \end{thm} In Section 3, we also show that for$H(r,s)=g(r)f(s)$, the following result holds. \begin{thm} \label{FKO} Let$H(r,s)=g(r)f(s)$satisfy (H1)--(H3), with$g(0)>0$and$g$positive, non-decreasing near$R$. Suppose (\ref{meqn}) has a blow-up solution$u$such that$u\in L^{(\alpha-p)/p}(0,R)$for some$\alpha>p$. If$g\in L^{1/\sigma}(0,R)$with$0<\sigma0$imply that$f(s)>0$for$s>0$. Since$f(s)>0$, it follows from (H1) that$g$is non-negative on$[0,R)$. \end{rem} Finally, we give some corollaries to Theorem \ref{FKO}. \section{A Comparison Result} We will need the following comparison lemma (see \cite{RAW} for a proof). For notational convenience in stating the lemma and in this section, we let$L$denote the differential operator on the left hand side of equation (\ref{meqnre}) above. In this lemma, we use the following notation:$u(a+)0$such that$u1$, then $$\lim_{t\to\infty}\frac{t^\alpha}{F(t)}=0.$$ \end{lem} We need the following lemma, which shows that solutions of (\ref{meqn}) with initial slope zero have non-decreasing slope for$r \in [0,R)$. \begin{lem}\label{prel} Suppose in addition to (H1)--(H3),$H(r,\cdot)$is non-decreasing on$[0,R)$. If for$0w_k(r)$for some$0 1/R$. For$t,r\in (0,R-1/k)$, and$n>k$we have $|w_n(r)-w_n(t)|=\big|\int_t^rw'_n(s)\,ds\big| \leq w'_n(\zeta)|r-t| \leq w'_{k+1}(R-1/k)|r-t|,$ where$\zeta=\max\{r,t\}$. The fact that$w'_{k+1}$is non-decreasing, by Lemma \ref{prel}, has been exploited in the last inequality. Thus$\{w_n\}_{n=k+1}^\infty$is a bounded equicontinuous family in$C([0,R-1/k])$, and hence has a uniformly convergent subsequence. Let$w$be the limit. For$r\in[0,R-1/k]$and$n>k$the solution$w_n$satisfies the integral equation $$w_n(r)=w_n(0)+\int_0^r\Big(\int_0^t\left(\frac{s}{t}\right)^\lambda H(0,w_n(s))\,ds\Big)^{1/(p-1)}\,dt\,.$$ Letting$n\to\infty$we see that$w$satisfies the same integral equation. Since$k$is arbitrary we conclude that$w$satisfies equation (\ref{seqn}). Since$u\leq w_n$on$(0,R-1/k)$for each$n\geq k$we conclude that$u\leq w$on$(0,R)$. \end{proof} \section{Proofs of Main Results and Some Corollaries} \begin{proof}[Proof of Theorem \ref{uint}] By Theorem \ref{scomp} we take a solution$w$of (\ref{seqn}) such that$u(r)\leq w(r)$for$0\leq r\frac{1}{\lambda+1}f(w),\quad 00$. Suppose (\ref{meqn}) has a blow-up solution that belongs to$L^{(\alpha-p)/p}(0,R)$for some$\alpha>p$. If$g$is bounded, then$F\in KO(\gamma)$for any$0<\gamma<\alpha$. \end{cor} \begin{rem} \label{rmk3.2}\rm The conclusion of Corollary \ref{coro3.1} is false when$g$is unbounded near$R$as the following example shows. The function$u(r)=(1-r)^{-1}$is a solution of \begin{gather*} u''(r)=g(r)f(u),\\ u(0)\geq 0,\quad u'(0)\geq 0,\quad u(1)=\infty, \end{gather*} where $$g(r):=2/(1-r),\quad \text{and}\quad f(s):=s^2.$$ Observe that$u\in L^{(\alpha-2)/2}(0,1)$for$2<\alpha<4$. However note that$F\notin KO(3)$. \end{rem} \begin{cor} \label{coro3.2} Suppose$H(r,s)=g(r)f(s)$satisfies (H1)--(H3), with$g(0)>0$,$g$non-decreasing on$[0,R)$, and$f(0)=0$. Further, let$g$be bounded on$[0,R)$, and let$F\in KO(p)$. Then a blow up solution$u$of (\ref{meqnre}) belongs to$L^q(0,R)$for some$q>0$if and only if$F\in KO(\gamma)$for some$\gamma>p$. \end{cor} \begin{proof} Suppose$F\in KO(\gamma)$for some$\gamma>p$. Then by Theorem \ref{uint}, we see that$u\in L^q(0,R)$for$q=(\gamma-p)/p$. For the converse, suppose that$u\in L^q(0,R)$for some$q>0$. Then for$\alpha=p(q+1)$we see that$q=(\alpha-p)/p$so that by the above corollary,$F\in KO(\gamma)$for some$p<\gamma